Mask Blank Substrate Polynomial Approximation for Position Shift Correction
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Solution Overview
Problem
The existing methods for manufacturing mask blanks in the semiconductor industry face challenges in accurately correcting transfer patterns due to the large amount of data required for simulating substrate shapes, which leads to difficulties in handling and processing, especially with the increasing need for higher positioning accuracy in double patterning techniques.
Innovation Solution
A method is introduced that involves approximating simulation results to a predetermined approximate curved surface, using a function of several variables in a three-dimensional coordinate system, and recording the coefficients of this surface to reduce data complexity and enhance calculation efficiency, allowing for accurate position shift corrections during the manufacturing process.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If simulation results are used as they are for correcting transfer patterns, then positioning accuracy is improved, but data handling becomes difficult due to large data amount
Solution Approach 1:
The patent extracts only the essential characteristics of the substrate shape from the complete simulation results. By fitting the simulation data to an approximate curved surface represented by a function of several variables, the method extracts the key positional information while discarding redundant data points, thereby reducing data handling complexity while preserving positioning accuracy.
Solution Approach 2:
The patent transforms the simulation results from a large set of discrete coordinate points into a compact mathematical model with a small number of parameters (coefficients of the approximate curved surface function). This parameter transformation maintains the essential positional information needed for accurate pattern correction while dramatically reducing the data volume that needs to be handled.
2Manufacturing precision
If complete simulation data is processed for pattern correction, then correction accuracy is improved, but calculation time increases
Solution Approach 1:
The patent extracts the essential shape characteristics of the substrate into a compact approximate curved surface model. This extraction process identifies and retains only the critical geometric information needed for accurate pattern correction, eliminating redundant computational data and thereby reducing calculation time while maintaining correction accuracy.
Solution Approach 2:
The patent creates a simplified mathematical copy (approximate curved surface function) that represents the complex simulation results. This copied model captures the essential substrate shape behavior with far fewer data points, enabling rapid calculations for pattern correction without requiring processing of the complete original simulation dataset.
3Measurement precision
If polynomial approximation of higher order is used, then approximation accuracy is improved, but calculation complexity increases
Solution Approach 1:
The patent systematically varies the order of the polynomial function to optimize the balance between approximation accuracy and calculation complexity. By adjusting the polynomial order parameter, the method achieves sufficient accuracy for practical applications while avoiding the excessive computational burden of unnecessarily high-order polynomials.
Solution Approach 2:
The patent employs a polynomial approximation order that is sufficient to capture the essential substrate shape characteristics but not excessively high. This partial action approach uses just enough mathematical complexity to achieve the required approximation accuracy without the diminishing returns and increased computational burden associated with higher-order polynomials.
Data Source
AI summary
In a simulation step of simulating a surface configuration of a substrate which is used for a mask blank and which is set to an exposure apparatus, height information from a reference plane is derived from a plurality of measurement points on a main surface of the substrate. From the height information, a curved surface of fourth, fifth, or sixth order is approximated which is represented by a polynomial specified by a plurality of terms and coefficients of the terms. The coefficients are stored as coefficient information in association with the substrate.